Math 2.0

Geometry Theorem

Tangent–secant theorem

Builds on Tangent line to a circleCircleEuclidean distancePower of a point

Statement

Let CC be a circle in the Euclidean plane ℝ2\mathbb{R}^2, and let dd be the Euclidean distance. Let PP be a point outside CC. Let a tangent line through PP touch CC at TT. Let a second line through PP meet CC at the two points G1G_1 and G2G_2. Then d(P,T)2=d(P,G1) d(P,G2).d(P, T)^2 = d(P, G_1)\, d(P, G_2).

Remarks

Both sides are equal to the power of PP with respect to CC. It is the limit case of the intersecting secants theorem where one secant becomes a tangent. The theorem is Proposition 36 of Book III of Euclid's Elements.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Circle, concept, no evidence levelCircleconceptEuclidean distance, concept, no evidence levelEuclidean distanceconceptPower of a point, concept, no evidence levelPower of a pointconceptTangent line to a circle, concept, no evidence levelTangent line to a cir…conceptTangent–secant theorem, statement, no evidence levelTangent–secant theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circle concept
  2. Euclidean distance concept
  3. Power of a point concept
  4. Tangent line to a circle concept
  5. Tangent–secant theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Tangent%E2%80%93secant_theorem, lead section
About this page
Tag0QX7E9KAAX
URIm2://0QX7E9KAAX
Kindstatement, theorem
WritingW1 stated
MSC51M04
StatusThis copy of the site has no build data. Run m2 status 0QX7E9KAAX.
UsesTangent line to a circle, Circle, Euclidean distance, Power of a point
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